Teaching Background and Step-by-Step

In finance, a num´eraire is any positive non-dividend paying asset. The following result, taken from [15] (refer to Proposition 2.2.1 in [15]), provides a fundamental tool for the pricing of derivatives to any num´eraire.

Proposition 2.2. [15] Assume there exists a num´eraire N and a probability measure Q^N, equivalent to the initial Q, such that the price of any traded asset X (without intermediate payments) relative to N is a martingale under Q^N, that is,

X(t)

N (t) = E^QN X(T ) N (T )   F (t)



, 0 ≤ t ≤ T. (2.21)

Let U be an arbitrary num´eraire. Then there exists a probability measure Q^U, equivalent to Q, such that the price of any attainable claim Y normalized by U is a martingale under Q^U, that is,

Y (t)

U (t) = E^QU Y (T ) U (T )   F (t)



, 0 ≤ t ≤ T. (2.22)

Moreover, the Radon-Nikod´ym derivative defined by Q^U is given by dQ^U

dQ^N = U (T )N (0) U (0)N (T ).

A zero-coupon bond with the maturity time T is a contract between two parties, namely the holder and the writer, that guarantees its holder the payment of one unit of currency at time T , with no intermediate payments. The contract value at time t < T is denoted by P (t, T ). Clearly, P (T, T ) = 1 for all T > 0.

Under the arbitrage-free assumption, we know that there is a risk-neutral probability measure Q for the market. Now, take a zero-coupon bond as the num´eraire U in Proposition 2.2, and let Q^T denote the corresponding probability measure, equivalent to Q. Realizing that P (T, T ) = 1 eliminates the dependence on the discount process, we obtain

Y (t) = P (t, T )E^T[Y (T )|F (t)], 0 ≤ t ≤ T. (2.23) We call Q^T the T -forward measure. It can be verified that the expectation of a future instantaneous spot rate r(T ) under Q^T is equal to the related instantaneous forward rate f (t, T ), that is,

E^T[r(T )|F (t)] = f (t, T ) for each 0 ≤ t ≤ T . For details, refer to [15].

In general, there is more complexity involved in pricing interest rate derivatives because the payoff functions depend on interest rates at multiple time points. Fur-thermore, the volatilities of these interest rates may differ due to the different peri-ods involved comprising from the short-term rates to the long-term rates. Whenever stochastic interest rates are present, there exists a joint dynamism between the under-lying asset price and interest rates in the pricing procedure. The use of the forward measure allows taking out the discounting effect from the joint evolution of the asset price and interest rates, when zero-coupon bonds are used as num´eraires.

A forward contract is an agreement between two parties, namely the holder and

the writer, where the holder agrees to buy an asset from the writer at delivery time T in the future for a pre-determined delivery price K. In this transaction, no up-front payment occurs. The delivery price is chosen so that the value of the forward contract to both parties is zero when the contract starts. The holder assumes a long position, and the writer assumes a short position.

A variance swap is a forward contract on the future realized variance of the returns of a specified asset. Offering additional purpose in determining the payoff of the financial derivative, this gives extra credit apart from the advantage of avoiding direct exposures to itself. Since the payment of a variance swap is only made in a single fixed payment at maturity, it is defined as a forward contract which is traded over the counter. At maturity time T , a variance swap rate can be evaluated as V (T ) = (RV −K)×L, where K is the annualized delivery or strike price for the variance swap and L is the notional amount of the swap in dollars. Roughly speaking, the realized variance (RV) is the sum of squared returns. It provides a relatively accurate measure of volatility which is useful for many purposes, including volatility forecasting and forecast evaluation.

The formula for the measure of realized variance used in this thesis and several other authors [80, 119] is

whereas in the market, a typical measure of the realized variance is defined as

RV = AF

The formula in (2.24) is known as the actual return realized variance, and the formula in (2.25) is recognizable as the log return realized variance. The formula in (2.25) had also been used extensively in the literature, such as in [116] and [117]. Several authors also used both definitions in their research, for example [34]. Here, S(tj) is the closing price of the underlying asset at the j -th observation time t_j, T is the lifetime of the contract and N is the number of observations. AF is the annualized factor which follows the sampling frequency to convert the above evaluation to annualized variance points. Assuming there are 252 business days in a year, AF = 252 for every trading day sampling frequency. However if the sampling frequency is every month or every week, then AF will be 12 and 52 respectively. The measure of realized variance requires

monitoring the underlying price path discretely, usually at the end of each business day.

For this purpose, we assume equally discrete observations to be compatible with the real market, which reduces to AF = _∆t¹ = ^N_T.

The long position of variance swaps pays a fixed delivery price at the expiration and receives the floating amounts of the annualized realized variance, whereas the short position is the opposite. The notional amount L can be expressed in two terms which are variance notional and vega notional. Variance notional gives the dollar amount of profit or loss obtained from the difference of one point between the realized variance and the delivery price. In contrast, vega notional calculates the profit or loss from one point of change in volatility points. Since it is the market practice to define the variance notional in volatility terms, the notional amount is typically quoted in dollars per volatility point. Even though the vega notional is the common market practice, this does not rise any complication due to the square-root relationship between the variance and volatility.

Generally, short position holders are mostly drawn to the irresistible attributes of variance swaps since the implied volatility is likely to be bigger than the final realized volatility. Moreover, the convexity property allows strike prices to be more expensive than the ones acquired from fair volatility. In addition, variance swaps also offer the capability to record the volatility trends of two correlated indices.